78 ANALYTICAL MECHANICS In order to express this definition of velocity in analytical language, consider a particle describing a, curved path with a changing speed. The most natural way of determining the speed at a point P, Fig- 47, is (o observe the interval of time which the particle takes to pass two points, I\ and P2, which are equidistant from P, \-then to divide the distance Pd\ by that interval of time. This gives the average speed from Pi to P2? which may or may not equal the actual speed at P. If, however, we take the points Pi and P<> nearer to P we obtain an average speed which is, in general, nearer the speed at P, because there is less chance for large variations. If we ° take Pi and P2 nearer and nearer the average speed approaches more and more to tho value ut P. Therefore the limiting value of the ratio \ " is the Therefore the limiting value of the ratio l " speed at P. In other words <** . * V^dt=* (I) is the analytical definition of speed. Therefore the velocity is a vector which has s for its magnitude* and which in tangent to the path at the point considered, that is^ V«8. (I') * The Differential Calculus was inventcnl by Newton and Leibnitz independently. Newton adopted a notation in which the derivative of a variable s with respect to another variable in denoted by £. Thin notation in not, eon-venient when derivatives are taken with respect to several variables. The notation introduced by Leibnitz ia more convenient and in the* notation which is generally adopted. Newton's notation, however, in often iinni to denote differentiation with respect to time. On account of the compactness of £ compared with jg, we will denote differential ion** with rc*i>cct to time by Newton's notation whenever compactness of expression is desired.